*Monday November 22*.

Please read the instructions for completing homeworks.

# Concepts

## Question 1

What is the difference between a **finitely** repeated game and an **infinitely** repeated game? What is the pure strategy Nash equilibrium in a finitely-repeated game (with a unique Nash equilibrium in a one-shot version)? Describe *two* interpretations of *infinitely* repeated games.

## Question 2

Describe, in your own words, the simple version (or implications) of the **folk theorem** for sustaining cooperation.

## Question 3

Describe what a **subgame** means, and circle all subgames in the following game tree.

## Question 4

Define **subgame perfect Nash equilibrium**.

## Question 5

Explain what **strategic moves** are, and explain the three major types of strategic moves.

## Question 6

What makes a *promise* **credible**? What makes a *threat* **credible**? Give some examples of each, and in your answers, use the concept of **subgame perfection.**

## Question 7

What makes a strategy **evolutionarily stable** (ESS)? Describe the difference between **monomorphic** and **polymorphic** equilibria.

# Problems

## Question 8

Consider an evolutionary version of the Stag Hunt game, where members of a species can decide to cooperate and hunt a **Stag** together, or defect and go after a **Hare** on their own.

### Part A

Is **Stag** an evolutionarily stable strategy (ESS)?

### Part B

Is **Hare** an evolutionarily stable strategy (ESS)?

### Part C

What are the pure strategy Nash equilibria (PSNE) of this game? Reconcile this with your answers in parts a and b.

### Part D

Suppose the environment changes such that hunting a large **Hare** *alone* is equally rewarding to the cooperative hunt of a **Stag** (but if they both hunt **Hare**, it is less rewarding).

Under the new environment, is **Hare** evolutionarily stable (ESS)?

### Part E

Under the new environment, is **Stag** evolutionarily stable (ESS)?

### Part F

Given what we learned in class about the relationship between (pure strategy) Nash equilibria and evolutionarily stable strategies, we now need a new refinement. Define a ** strict Nash equilibrium** in pure strategies to mean that each player is playing a

**strict**(or

**unique**) best response to other players, i.e. there is no

*other*strategy that is

*also*a best response to another player. In the one-shot game in part d, which PSNE are

**strict**, and which are not (i.e. “weak” PSNE? What do you then think is the relationship between ESS and strict/non-strict PSNE?

## Question 9

Consider the evolutionary Hawk-Dove game, where members of a species are competing over a resource valued at 12, with a cost of losing a fight being \(-15\).

### Part A

Draw the payoff matrix for the game.

### Part B

Find the pure strategy Nash equilibria.

### Part C

Is **Hawk** evolutionarily stable?

### Part D

Is **Dove** evolutionarily stable?

### Part E

Reconcile your answers in parts c and d to your answer in part b.

### Part F

Find the evolutionarily stable (polymorphic) equilibrium distribution of **Hawks** and **Doves.** [Hint: let \(p\) be the probability the *other* player is a **Hawk**.]

## Question 10

Consider the following game between two roommates. Roommate A has a very difficult exam the next morning, while Roommate B does not. The two of them can each decide to Study or Go Out that evening. Both would rather do something together, while A would certainly prefer they both Study and B would prefer they both Go Out.

### Part A

Suppose they both agree that A gets to decide first and B must respond, as in the following game:

Solve this game for the rollback equilibrium using backwards induction.

### Part B

Circle all subgames on the game tree.

### Part C

Carefully convert this game from extensive form to strategic form. (Be mindful of how many potential strategies each player has!) Then, find any Nash equilibria in strategic form.

### Part D

Which Nash equilibrium is subgame perfect? Why?

### Part E

Suppose in an attempt to get A to Go Out, B says they will Go Out regardless of what A does. If A still gets to decide first (i.e. it is the same game as in part a), what should A make of this?